Wave scattering from periodic arrays is ubiquitous in applied mathematics, and has received a great deal of attention over the past century, not least due to the physical significance of understanding the reflection and transmission of plane waves from such arrays in the contexts of electromagnetic waves, acoustics, water waves and elasticity. The aim of the thesis is to develop an accurate and efficient numerical method to solve for the reflection and transmission of an acoustic plane wave from arrays of arbitrary shaped obstacles that have an axis of symmetry aligned in a direction perpendicular to the array. We are particularly interested in the difficult case when the characteristic length scale of the scatterers, and the periodic spacing of the array are of the same order of magnitude as the wavelength of the incident wave. It is shown that the boundary value problem for the infinite array can be reduced to an integral equation over a central representative cell containing a single scatterer, which can then be solved using the boundary element method. Particular attention is paid to the convergence of the resulting periodic Green's function. Using established methods to calculate the reflection and transmission coefficients, we develop a new method to increase the rate of convergence of the periodic Green's function in both two and three dimensions.